Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander - - PowerPoint PPT Presentation

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Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander - - PowerPoint PPT Presentation

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Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander Verdonschot Carleton University 10 August 2015 Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations Pointed: all vertices are incident to a

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SLIDE 1

Flips in Edge-Labelled Pseudo-Triangulations

Prosenjit Bose Sander Verdonschot

Carleton University

10 August 2015

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 2

Pseudo-triangulations

  • All faces are pseudo-triangles
  • Pointed: all vertices are incident to a reflex angle (

)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 3

Pseudo-triangulations

  • All faces are pseudo-triangles
  • Pointed: all vertices are incident to a reflex angle (> π)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 4

Pseudo-triangulations

  • All faces are pseudo-triangles
  • Pointed: all vertices are incident to a reflex angle (> π)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 5

Flips

  • Remove edge, leaving a pseudo-quadrilateral
  • Find corners opposite removed edge
  • Insert connecting geodesic

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 6

Previous work

.

Theorem (Bereg, 2004)

. . Any pointed pseudo-triangulation can be transformed into any

  • ther with O(n log n) flips.
  • What happens when edges are labelled?

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 7

Previous work

.

Theorem (Bereg, 2004)

. . Any pointed pseudo-triangulation can be transformed into any

  • ther with O(n log n) flips.
  • What happens when edges are labelled?

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 8

Upper bound

  • Transform T1 into T2
  • Via canonical form TC
  • We only need to show T

TC

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 9

Upper bound

  • Transform T1 into T2
  • Via canonical form TC
  • We only need to show T

TC

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 10

Upper bound

  • Transform T1 into T2
  • Via canonical form TC
  • We only need to show T → TC

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 11

Transform into canonical

  • Ignore labels
  • Move labels around

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 12

Left-shelling pseudo-triangulation

  • Add vertices in clockwise order around bottom vertex
  • Connect to bottom (bottom edge)
  • Add tangent to convex hull (top edge)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 13

Tools

  • Sweep: exchange labels on top and bottom pairs
  • Shuffle: reorder bottom labels

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 14

Algorithm

  • Identify out-of-place top and bottom labels

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 15

Algorithm

  • Pair these up (Shuffle)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 16

Algorithm

  • Exchange them (Sweep)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 17

Algorithm

  • Sort bottom labels (Shuffle)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 18

Algorithm

  • Sort bottom labels (Shuffle)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 19

Algorithm

  • Move all top labels down (Sweep)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 20

Algorithm

  • Move all top labels down (Sweep)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 21

Algorithm

  • Sort them (Shuffle)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 22

Algorithm

  • Move them back (Sweep)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 23

Upper bound

.

Theorem

. . We can sort the labels of a left-shelling pseudo-triangulation with O(1) shuffles and sweeps.

  • How do we shuffle and sweep?

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 24

Upper bound

.

Theorem

. . We can sort the labels of a left-shelling pseudo-triangulation with O(1) shuffles and sweeps.

  • How do we shuffle and sweep?

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 25

Sweep

  • Easy for degree-2 vertices:
  • Idea: make every vertex degree-2 at some point

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 26

Sweep

  • Shoot a ray from vbottom to the right

When it passes a vertex:

  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 27

Sweep

  • Sweep it counter-clockwise through the point set

When it passes a vertex:

  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 28

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 29

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 30

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 31

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 32

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 33

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 34

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 35

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 36

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 37

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 38

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 39

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 40

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 41

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 42

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 43

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 44

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 45

Sweep

  • When it passes a vertex:
  • Swap the top and bottom edge, if necessary
  • Flip the top edge

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 46

Tools

  • Sweep: exchange labels on top and bottom pairs — O(n)
  • Shuffle: reorder bottom labels

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 47

Shuffle

  • Swap consecutive bottom edges
  • Easy if third pseudo-triangle is a triangle
  • Otherwise, flip top edge first
  • We can do insertion sort!

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 48

Shuffle

  • Swap consecutive bottom edges
  • Easy if third pseudo-triangle is a triangle
  • Otherwise, flip top edge first
  • We can do insertion sort!

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 49

Shuffle

  • Swap consecutive bottom edges
  • Easy if third pseudo-triangle is a triangle
  • Otherwise, flip top edge first
  • We can do insertion sort!

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 50

Tools

  • Sweep: exchange labels on top and bottom pairs — O(n)
  • Shuffle: reorder bottom labels — O(n2)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 51

Upper bound

.

Theorem

. . We can sort the labels of a left-shelling pseudo-triangulation with O(1) shuffles and sweeps. .

Theorem

. . We can transform any edge-labelled pointed pseudo-triangulation into any other with O(n2) flips.

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 52

All pseudo-triangulations

  • Number of edges differs
  • We need edge-insertion and edge-deletion flips
  • Edge label is drawn from / placed into pool of free labels

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 53

All pseudo-triangulations

  • Number of edges differs
  • We need edge-insertion and edge-deletion flips
  • Edge label is drawn from / placed into pool of free labels

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 54

All pseudo-triangulations

  • Swap edges incident to degree-2 vertices

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 55

All pseudo-triangulations

  • Swap edges incident to degree-2 vertices

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 56

All pseudo-triangulations

  • Swap edges incident to degree-2 vertices

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 57

All pseudo-triangulations

  • Swap edges incident to degree-2 vertices

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 58

All pseudo-triangulations

  • Swap edges incident to degree-2 vertices

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 59

Upper bound

.

Theorem

. . Using insertion and deletion flips, we can shuffle with O(n log n) flips. .

Theorem

. . We can transform any edge-labelled pseudo-triangulation into any other with O(n log n) flips.

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 60

Lower bound

  • Triangulation of convex polygon

= pointed pseudo-triangulation .

Theorem

. . There are pairs of edge-labelled pointed pseudo-triangulations such that we need at least Ω(n log n) flips to transform one into the other.

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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SLIDE 61

Summary

  • Pointed pseudo-triangulations (only exchanging flips)

Ω(n log n) and O(n2)

  • All pseudo-triangulations (all three flip types)

Θ(n log n)

Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations